Fit log returns to F-S skew standardized Student-t
distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of
freedom).
xi is the estimated skewness parameter.
For 2011, medium risk data is used in the high risk data set, as no
high risk fund data is available prior to 2012.
vmrl is a long version of Velliv medium risk data, from
2007 to 2023. For 2007 to 2011 (both included) no high risk data is
available.
| vmr | vhr | vmrl | pmr | phr | mmr | mhr | vm_ph_r | vh_pm_r | |
|---|---|---|---|---|---|---|---|---|---|
| Min. : | 0.868 | 0.849 | 0.801 | 0.904 | 0.878 | 0.988 | 0.977 | 0.979 | 0.967 |
| 1st Qu.: | 1.044 | 1.039 | 1.013 | 1.042 | 1.068 | 1.013 | 1.013 | 1.021 | 1.012 |
| Median : | 1.097 | 1.099 | 1.085 | 1.084 | 1.128 | 1.085 | 1.113 | 1.102 | 1.094 |
| Mean : | 1.070 | 1.085 | 1.061 | 1.065 | 1.095 | 1.066 | 1.087 | 1.081 | 1.074 |
| 3rd Qu.: | 1.136 | 1.160 | 1.128 | 1.107 | 1.182 | 1.101 | 1.128 | 1.121 | 1.106 |
| Max. : | 1.168 | 1.214 | 1.193 | 1.141 | 1.208 | 1.133 | 1.207 | 1.178 | 1.163 |
| Min. : | ranking | 1st Qu.: | ranking | Median : | ranking | Mean : | ranking | 3rd Qu.: | ranking | Max. : | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.988 | mmr | 1.068 | phr | 1.128 | phr | 1.095 | phr | 1.182 | phr | 1.214 | vhr |
| 0.979 | vm_ph_r | 1.044 | vmr | 1.113 | mhr | 1.087 | mhr | 1.160 | vhr | 1.208 | phr |
| 0.977 | mhr | 1.042 | pmr | 1.102 | vm_ph_r | 1.085 | vhr | 1.136 | vmr | 1.207 | mhr |
| 0.967 | vh_pm_r | 1.039 | vhr | 1.099 | vhr | 1.081 | vm_ph_r | 1.128 | vmrl | 1.193 | vmrl |
| 0.904 | pmr | 1.021 | vm_ph_r | 1.097 | vmr | 1.074 | vh_pm_r | 1.128 | mhr | 1.178 | vm_ph_r |
| 0.878 | phr | 1.013 | vmrl | 1.094 | vh_pm_r | 1.070 | vmr | 1.121 | vm_ph_r | 1.168 | vmr |
| 0.868 | vmr | 1.013 | mmr | 1.085 | vmrl | 1.066 | mmr | 1.107 | pmr | 1.163 | vh_pm_r |
| 0.849 | vhr | 1.013 | mhr | 1.085 | mmr | 1.065 | pmr | 1.106 | vh_pm_r | 1.141 | pmr |
| 0.801 | vmrl | 1.012 | vh_pm_r | 1.084 | pmr | 1.061 | vmrl | 1.101 | mmr | 1.133 | mmr |
Correlations
| vmr | vhr | pmr | phr | |
|---|---|---|---|---|
| vmr | 1.000 | 0.993 | -0.197 | -0.095 |
| vhr | 0.993 | 1.000 | -0.119 | -0.016 |
| pmr | -0.197 | -0.119 | 1.000 | 0.957 |
| phr | -0.095 | -0.016 | 0.957 | 1.000 |
Covariances
| vmr | vhr | pmr | phr | |
|---|---|---|---|---|
| vmr | 0.007 | 0.009 | -0.001 | -0.001 |
| vhr | 0.009 | 0.011 | -0.001 | 0.000 |
| pmr | -0.001 | -0.001 | 0.004 | 0.007 |
| phr | -0.001 | 0.000 | 0.007 | 0.011 |
Risk of max loss of x percent for a single period
(year).
x values are row names.
| vmr | vhr | pmr | phr | mmr | mhr | vm_ph_r | vh_pm_r | |
|---|---|---|---|---|---|---|---|---|
| 0 | 21.167 | 21.333 | 11.833 | 14.000 | 12.333 | 12.667 | 16.667 | 16.000 |
| 5 | 12.167 | 13.167 | 5.667 | 8.333 | 5.833 | 3.833 | 8.667 | 8.167 |
| 10 | 7.000 | 8.000 | 3.000 | 5.000 | 2.833 | 0.500 | 4.333 | 4.167 |
| 25 | 1.333 | 1.500 | 0.500 | 1.000 | 0.333 | 0.000 | 0.333 | 0.333 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 90 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 99 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 21.333 | vhr | 13.167 | vhr | 8.000 | vhr | 1.500 | vhr | 0 | vmr | 0 | vmr | 0 | vmr |
| 21.167 | vmr | 12.167 | vmr | 7.000 | vmr | 1.333 | vmr | 0 | vhr | 0 | vhr | 0 | vhr |
| 16.667 | vm_ph_r | 8.667 | vm_ph_r | 5.000 | phr | 1.000 | phr | 0 | pmr | 0 | pmr | 0 | pmr |
| 16.000 | vh_pm_r | 8.333 | phr | 4.333 | vm_ph_r | 0.500 | pmr | 0 | phr | 0 | phr | 0 | phr |
| 14.000 | phr | 8.167 | vh_pm_r | 4.167 | vh_pm_r | 0.333 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 12.667 | mhr | 5.833 | mmr | 3.000 | pmr | 0.333 | vm_ph_r | 0 | mhr | 0 | mhr | 0 | mhr |
| 12.333 | mmr | 5.667 | pmr | 2.833 | mmr | 0.333 | vh_pm_r | 0 | vm_ph_r | 0 | vm_ph_r | 0 | vm_ph_r |
| 11.833 | pmr | 3.833 | mhr | 0.500 | mhr | 0.000 | mhr | 0 | vh_pm_r | 0 | vh_pm_r | 0 | vh_pm_r |
Chance of min gains of x percent for a single period
(year).
x values are row names.
| vmr | vhr | pmr | phr | mmr | mhr | vm_ph_r | vh_pm_r | |
|---|---|---|---|---|---|---|---|---|
| 0 | 78.833 | 78.667 | 88.167 | 86.000 | 87.667 | 87.333 | 83.333 | 84.000 |
| 5 | 63.833 | 66.667 | 71.667 | 76.000 | 71.667 | 70.167 | 69.333 | 69.000 |
| 10 | 40.833 | 50.167 | 32.500 | 59.667 | 35.500 | 46.000 | 47.167 | 43.833 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.833 | 0.000 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 100 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 88.167 | pmr | 76.000 | phr | 59.667 | phr | 0.833 | mhr | 0 | vmr | 0 | vmr |
| 87.667 | mmr | 71.667 | pmr | 50.167 | vhr | 0.000 | vmr | 0 | vhr | 0 | vhr |
| 87.333 | mhr | 71.667 | mmr | 47.167 | vm_ph_r | 0.000 | vhr | 0 | pmr | 0 | pmr |
| 86.000 | phr | 70.167 | mhr | 46.000 | mhr | 0.000 | pmr | 0 | phr | 0 | phr |
| 84.000 | vh_pm_r | 69.333 | vm_ph_r | 43.833 | vh_pm_r | 0.000 | phr | 0 | mmr | 0 | mmr |
| 83.333 | vm_ph_r | 69.000 | vh_pm_r | 40.833 | vmr | 0.000 | mmr | 0 | mhr | 0 | mhr |
| 78.833 | vmr | 66.667 | vhr | 35.500 | mmr | 0.000 | vm_ph_r | 0 | vm_ph_r | 0 | vm_ph_r |
| 78.667 | vhr | 63.833 | vmr | 32.500 | pmr | 0.000 | vh_pm_r | 0 | vh_pm_r | 0 | vh_pm_r |
Risk of loss from first to last period.
_m is medium.
_h is high.
a is simulation from estimated distribution of returns
of mix.
b is mix of simulations from estimated distribution of
returns from individual funds.
l for “long”, going back to 2007.
| vmr | vhr | pmr | phr | mmr | mhr | vm_ph_r | vh_pm_r | |
|---|---|---|---|---|---|---|---|---|
| 0 | 4.91 | 2.65 | 1.92 | 0.98 | 1.18 | 0 | 0.63 | 0.69 |
| 5 | 4.32 | 2.37 | 1.68 | 0.88 | 1.04 | 0 | 0.56 | 0.55 |
| 10 | 3.67 | 2.04 | 1.47 | 0.80 | 0.90 | 0 | 0.44 | 0.47 |
| 25 | 2.33 | 1.16 | 1.09 | 0.56 | 0.55 | 0 | 0.22 | 0.27 |
| 50 | 0.82 | 0.38 | 0.63 | 0.33 | 0.24 | 0 | 0.04 | 0.13 |
| 90 | 0.05 | 0.02 | 0.14 | 0.07 | 0.03 | 0 | 0.00 | 0.02 |
| 99 | 0.00 | 0.00 | 0.05 | 0.01 | 0.00 | 0 | 0.00 | 0.00 |
1e6 simulation paths of mhr_b:
| 0 | 5 | 10 | 25 | 50 | 90 | 99 | |
|---|---|---|---|---|---|---|---|
| prob_pct | 0.118 | 0.095 | 0.076 | 0.036 | 0.008 | 0 | 0 |
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.91 | vmr | 4.32 | vmr | 3.67 | vmr | 2.33 | vmr | 0.82 | vmr | 0.14 | pmr | 0.05 | pmr |
| 2.65 | vhr | 2.37 | vhr | 2.04 | vhr | 1.16 | vhr | 0.63 | pmr | 0.07 | phr | 0.01 | phr |
| 1.92 | pmr | 1.68 | pmr | 1.47 | pmr | 1.09 | pmr | 0.38 | vhr | 0.05 | vmr | 0.00 | vmr |
| 1.18 | mmr | 1.04 | mmr | 0.90 | mmr | 0.56 | phr | 0.33 | phr | 0.03 | mmr | 0.00 | vhr |
| 0.98 | phr | 0.88 | phr | 0.80 | phr | 0.55 | mmr | 0.24 | mmr | 0.02 | vhr | 0.00 | mmr |
| 0.69 | vh_pm_r | 0.56 | vm_ph_r | 0.47 | vh_pm_r | 0.27 | vh_pm_r | 0.13 | vh_pm_r | 0.02 | vh_pm_r | 0.00 | mhr |
| 0.63 | vm_ph_r | 0.55 | vh_pm_r | 0.44 | vm_ph_r | 0.22 | vm_ph_r | 0.04 | vm_ph_r | 0.00 | mhr | 0.00 | vm_ph_r |
| 0.00 | mhr | 0.00 | mhr | 0.00 | mhr | 0.00 | mhr | 0.00 | mhr | 0.00 | vm_ph_r | 0.00 | vh_pm_r |
Chance of gains from first to last period.
_a is simulation from estimated distribution of returns of
mix.
_b is mix of simulations from estimated distribution of
returns from individual funds.
| vmr | vhr | pmr | phr | mmr | mhr | vm_ph_r | vh_pm_r | |
|---|---|---|---|---|---|---|---|---|
| 0 | 95.09 | 97.35 | 98.08 | 99.02 | 98.82 | 100.00 | 99.37 | 99.31 |
| 5 | 94.49 | 96.90 | 97.89 | 98.85 | 98.63 | 100.00 | 99.29 | 99.16 |
| 10 | 93.80 | 96.46 | 97.75 | 98.63 | 98.45 | 100.00 | 99.16 | 98.99 |
| 25 | 91.24 | 95.08 | 97.10 | 98.27 | 97.63 | 100.00 | 98.60 | 98.41 |
| 50 | 85.83 | 92.13 | 95.35 | 97.17 | 95.88 | 99.99 | 97.39 | 96.62 |
| 100 | 71.80 | 83.79 | 88.74 | 94.41 | 89.61 | 99.59 | 91.79 | 90.50 |
| 200 | 39.30 | 61.73 | 59.83 | 85.16 | 59.41 | 92.88 | 69.79 | 64.10 |
| 300 | 16.30 | 39.66 | 23.06 | 71.21 | 22.97 | 71.44 | 41.76 | 32.81 |
| 400 | 5.33 | 23.14 | 4.29 | 54.62 | 3.78 | 43.88 | 19.93 | 12.51 |
| 500 | 1.44 | 12.68 | 0.48 | 38.57 | 0.25 | 22.42 | 7.79 | 3.76 |
| 1000 | 0.00 | 0.26 | 0.02 | 2.34 | 0.00 | 0.23 | 0.01 | 0.00 |
1e6 simulation paths of mhr_b:
| 0 | 5 | 10 | 25 | 50 | 100 | 200 | 300 | 400 | 500 | 1000 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| prob | 99.882 | 99.854 | 99.824 | 99.686 | 99.301 | 97.513 | 86.912 | 65.992 | 41.486 | 21.693 | 0.086 |
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 100.00 | mhr | 100.00 | mhr | 100.00 | mhr | 100.00 | mhr | 99.99 | mhr | 99.59 | mhr |
| 99.37 | vm_ph_r | 99.29 | vm_ph_r | 99.16 | vm_ph_r | 98.60 | vm_ph_r | 97.39 | vm_ph_r | 94.41 | phr |
| 99.31 | vh_pm_r | 99.16 | vh_pm_r | 98.99 | vh_pm_r | 98.41 | vh_pm_r | 97.17 | phr | 91.79 | vm_ph_r |
| 99.02 | phr | 98.85 | phr | 98.63 | phr | 98.27 | phr | 96.62 | vh_pm_r | 90.50 | vh_pm_r |
| 98.82 | mmr | 98.63 | mmr | 98.45 | mmr | 97.63 | mmr | 95.88 | mmr | 89.61 | mmr |
| 98.08 | pmr | 97.89 | pmr | 97.75 | pmr | 97.10 | pmr | 95.35 | pmr | 88.74 | pmr |
| 97.35 | vhr | 96.90 | vhr | 96.46 | vhr | 95.08 | vhr | 92.13 | vhr | 83.79 | vhr |
| 95.09 | vmr | 94.49 | vmr | 93.80 | vmr | 91.24 | vmr | 85.83 | vmr | 71.80 | vmr |
| 200 | ranking | 300 | ranking | 400 | ranking | 500 | ranking | 1000 | ranking |
|---|---|---|---|---|---|---|---|---|---|
| 92.88 | mhr | 71.44 | mhr | 54.62 | phr | 38.57 | phr | 2.34 | phr |
| 85.16 | phr | 71.21 | phr | 43.88 | mhr | 22.42 | mhr | 0.26 | vhr |
| 69.79 | vm_ph_r | 41.76 | vm_ph_r | 23.14 | vhr | 12.68 | vhr | 0.23 | mhr |
| 64.10 | vh_pm_r | 39.66 | vhr | 19.93 | vm_ph_r | 7.79 | vm_ph_r | 0.02 | pmr |
| 61.73 | vhr | 32.81 | vh_pm_r | 12.51 | vh_pm_r | 3.76 | vh_pm_r | 0.01 | vm_ph_r |
| 59.83 | pmr | 23.06 | pmr | 5.33 | vmr | 1.44 | vmr | 0.00 | vmr |
| 59.41 | mmr | 22.97 | mmr | 4.29 | pmr | 0.48 | pmr | 0.00 | mmr |
| 39.30 | vmr | 16.30 | vmr | 3.78 | mmr | 0.25 | mmr | 0.00 | vh_pm_r |
Summary for fit of log returns to an F-S skew standardized Student-t
distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape
parameter.
xi is the estimated skewness parameter.
| vmr | vhr | pmr | phr | mmr | mhr | vm_ph_r | vh_pm_r | |
|---|---|---|---|---|---|---|---|---|
| m | 0.048 | 0.063 | 0.058 | 0.084 | 0.059 | 0.082 | 0.067 | 0.062 |
| s | 0.120 | 0.126 | 0.123 | 0.121 | 0.088 | 0.071 | 0.091 | 0.090 |
| nu | 3.304 | 4.390 | 2.265 | 3.185 | 2.773 | 89.863 | 4.660 | 3.892 |
| xi | 0.034 | 0.019 | 0.477 | 0.018 | 0.029 | 0.770 | 0.048 | 0.019 |
| R^2 | 0.993 | 0.995 | 0.991 | 0.964 | 0.890 | 0.961 | 0.927 | 0.933 |
| m | ranking | s | ranking | R^2 | ranking |
|---|---|---|---|---|---|
| 0.084 | phr | 0.071 | mhr | 0.995 | vhr |
| 0.082 | mhr | 0.088 | mmr | 0.993 | vmr |
| 0.067 | vm_ph_r | 0.090 | vh_pm_r | 0.991 | pmr |
| 0.063 | vhr | 0.091 | vm_ph_r | 0.964 | phr |
| 0.062 | vh_pm_r | 0.120 | vmr | 0.961 | mhr |
| 0.059 | mmr | 0.121 | phr | 0.933 | vh_pm_r |
| 0.058 | pmr | 0.123 | pmr | 0.927 | vm_ph_r |
| 0.048 | vmr | 0.126 | vhr | 0.890 | mmr |
Monte Carlo simulations of portfolio index values (currency
values).
Statistics are given for the final state of all paths.
Probability of down-and_out is calculated as the share of paths that
reach 0 at some point. All subsequent values for a path are set to 0, if
the path reaches at any point.
0 is defined as any value below a threshold.
dai_pct (for down-and-in) is the probability of losing
money. This is calculated as the share of paths finishing below index
100.
## Number of paths: 10000
| vmr | vhr | pmr | phr | mmr | mhr | vm_ph_r | vh_pm_r | |
|---|---|---|---|---|---|---|---|---|
| mc_m | 295.32 | 409.52 | 345.08 | 601.31 | 345.65 | 541.75 | 411.22 | 378.07 |
| mc_s | 135.03 | 210.06 | 116.43 | 272.65 | 108.44 | 173.87 | 156.51 | 137.95 |
| mc_min | 4.82 | 2.88 | 0.00 | 1.04 | 1.45 | 160.90 | 34.17 | 4.87 |
| mc_max | 947.71 | 1593.81 | 2517.59 | 2247.55 | 748.29 | 1459.68 | 1230.60 | 1069.74 |
| dao_pct | 0.00 | 0.00 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| dai_pct | 4.44 | 2.39 | 1.73 | 0.89 | 1.03 | 0.00 | 0.53 | 0.62 |
| mc_m | ranking | mc_s | ranking | mc_min | ranking | mc_max | ranking | dao_pct | ranking | dai_pct | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 601.31 | phr | 108.44 | mmr | 160.90 | mhr | 2517.59 | pmr | 0.00 | vmr | 0.00 | mhr |
| 541.75 | mhr | 116.43 | pmr | 34.17 | vm_ph_r | 2247.55 | phr | 0.00 | vhr | 0.53 | vm_ph_r |
| 411.22 | vm_ph_r | 135.03 | vmr | 4.87 | vh_pm_r | 1593.81 | vhr | 0.00 | phr | 0.62 | vh_pm_r |
| 409.52 | vhr | 137.95 | vh_pm_r | 4.82 | vmr | 1459.68 | mhr | 0.00 | mmr | 0.89 | phr |
| 378.07 | vh_pm_r | 156.51 | vm_ph_r | 2.88 | vhr | 1230.60 | vm_ph_r | 0.00 | mhr | 1.03 | mmr |
| 345.65 | mmr | 173.87 | mhr | 1.45 | mmr | 1069.74 | vh_pm_r | 0.00 | vm_ph_r | 1.73 | pmr |
| 345.08 | pmr | 210.06 | vhr | 1.04 | phr | 947.71 | vmr | 0.00 | vh_pm_r | 2.39 | vhr |
| 295.32 | vmr | 272.65 | phr | 0.00 | pmr | 748.29 | mmr | 0.03 | pmr | 4.44 | vmr |
| vmr | vhr | pmr | phr | mmr | mhr | vm_ph_r | vh_pm_r | |
|---|---|---|---|---|---|---|---|---|
| m | 0.064 | 0.077 | 0.061 | 0.085 | 0.062 | 0.081 | 0.076 | 0.069 |
| s | 0.081 | 0.099 | 0.063 | 0.101 | 0.048 | 0.070 | 0.062 | 0.060 |
Probability in percent that the smallest and largest (respectively) observed return for each fund was generated by a normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vm_ph_r | vh_pm_r | |
|---|---|---|---|---|---|---|---|---|
| P_norm(X_min) | 0.571 | 0.758 | 0.511 | 1.676 | 5.971 | 6.842 | 5.945 | 4.228 |
| P_norm(X_max) | 13.230 | 11.876 | 12.922 | 15.359 | 9.628 | 6.429 | 7.796 | 8.592 |
| P_t(X_min) | 5.377 | 5.080 | 3.489 | 4.315 | 10.570 | 8.015 | 13.008 | 10.520 |
| P_t(X_max) | 0.118 | 0.156 | 2.825 | 0.188 | 0.488 | 5.141 | 0.229 | 0.175 |
Average number of years between min or max events (respectively):
| vmr | vhr | pmr | phr | mmr | mhr | vm_ph_r | vh_pm_r | |
|---|---|---|---|---|---|---|---|---|
| norm: avg yrs btw min | 175.248 | 131.911 | 195.568 | 59.669 | 16.748 | 14.616 | 16.820 | 23.650 |
| norm: avg yrs btw max | 7.559 | 8.420 | 7.739 | 6.511 | 10.386 | 15.556 | 12.827 | 11.639 |
| t: avg yrs btw min | 18.596 | 19.687 | 28.663 | 23.173 | 9.461 | 12.476 | 7.688 | 9.506 |
| t: avg yrs btw max | 848.548 | 640.410 | 35.400 | 531.552 | 205.104 | 19.450 | 437.280 | 572.483 |
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.2262221 0.3361598
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.5098519 0.4248085
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.2965857 0.3073935
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.8351623 0.4382935
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.2921756 0.3005586
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.7011721 0.3095095
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.4606802 0.3586853
Objective function plots
Let’s plot the fit and the observed returns together.
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Max vs sum plots for the first four moments:
Parameters
## [1] 1.3815768 0.3410388
Objective function plots
1e6 paths:
# Down-and-out simulation:
# Probability of down-and-out: 0 percent
#
# Mean portfolio index value after 20 years: 478.339 kr.
# SD of portfolio index value after 20 years: 163.093 kr.
# Min total portfolio index value after 20 years: 2.233 kr.
# Max total portfolio index value after 20 years: 1561.965 kr.
#
# Share of paths finishing below 100: 0.1181 percent
\[\text{Avg. of returns} := \dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2}\] \[\text{Returns of avg.} := \left(\dfrac{ x_t + y_t }{2}\right) \Big/ \left(\dfrac{ x_{t-1} + y_{t-1} }{2}\right) \equiv \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]
For which \(x_1\) and \(y_1\) are \(\text{Avg. of returns} = \text{Returns of avg.}\)?
\[\dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2} = \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]
\[\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]
\[(x_{t-1} + y_{t-1}) x_t y_{t-1} + (x_{t-1} + y_{t-1}) x_{t-1} y_t = 2 (x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\]
\[(x_{t-1}x_1y_{t-1} + y_{t-1}x_ty_{t-1}) + (x_{t-1}x_{t-1}y_t + x_{t-1}y_{t-1}y_t) = 2(x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\] This is not generally true, but true if for instance \(x_{t-1} = y_{t-1}\).
Definition: R = 1+r
## Let x_0 be 100.
## Let y_0 be 200.
## So the initial value of the pf is 300 .
## Let R_x be 0.5.
## Let R_y be 1.5.
Then,
## x_1 is R_x * x_0 = 50.
## y_1 is R_y * y_0 = 300.
Average of returns:
## 0.5 * (R_x + R_y) = 1
So here the value of the pf at t=1 should be unchanged from t=0:
## (x_0 + y_0) * 0.5 * (R_x + R_y) = 300
But this is clearly not the case:
## 0.5 * (x_1 + y_1) = 0.5 * (R_x * x_0 + R_y * y_0) = 175
Therefore we should take returns of average, not average of returns!
Let’s take the average of log returns instead:
## 0.5 * (log(R_x) + log(R_y)) = -0.143841
We now get:
## (x_0 + y_0) * exp(0.5 * (log(Rx) + log(Ry))) = 259.8076
So taking the average of log returns doesn’t work either.
Test if a simulation of a mix (average) of two returns series has the same distribution as a mix of two simulated returns series.
## m(data_x): 0.02798047
## s(data_x): 0.4736139
## m(data_y): 10.00934
## s(data_y): 3.435938
##
## m(data_x + data_y): 5.018661
## s(data_x + data_y): 1.861301
m and s of final state of all paths.
_a is mix of simulated returns.
_b is simulated mixed returns.
| m_a | m_b | s_a | s_b |
|---|---|---|---|
| 100.395 | 100.523 | 7.828 | 8.207 |
| 100.280 | 100.324 | 7.876 | 8.423 |
| 100.678 | 100.278 | 7.797 | 8.348 |
| 100.436 | 100.097 | 7.717 | 8.076 |
| 100.251 | 100.164 | 7.579 | 8.266 |
| 100.202 | 100.174 | 7.870 | 8.530 |
| 100.682 | 100.875 | 7.833 | 8.249 |
| 100.401 | 100.435 | 7.438 | 8.372 |
| 99.944 | 100.607 | 7.766 | 8.203 |
| 100.409 | 100.310 | 7.787 | 8.231 |
## m_a m_b s_a s_b
## Min. : 99.94 Min. :100.1 Min. :7.438 Min. :8.076
## 1st Qu.:100.26 1st Qu.:100.2 1st Qu.:7.730 1st Qu.:8.213
## Median :100.40 Median :100.3 Median :7.792 Median :8.258
## Mean :100.37 Mean :100.4 Mean :7.749 Mean :8.291
## 3rd Qu.:100.43 3rd Qu.:100.5 3rd Qu.:7.831 3rd Qu.:8.366
## Max. :100.68 Max. :100.9 Max. :7.876 Max. :8.530
_a and _b are very close to equal.
We attribute the differences to differences in estimating the
distributions in version a and b.
The final state is independent of the order of the preceding steps:
So does the order of the steps in the two processes matter, when mixing simulated returns?
The order of steps in the individual paths do not matter, because the mix of simulated paths is a sum of a sum, so the order of terms doesn’t affect the sum. If there is variation it is because the sets preceding steps are not the same. For instance, the steps between step 1 and 60 in the plot above are not the same for the two lines.
Recall, \[\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(a, b)\]
var(0.5 * vhr + 0.5 * phr)
## [1] 0.005355618
0.5^2 * var(vhr) + 0.5^2 * var(phr) + 2 * 0.5 * 0.5 * cov(vhr, phr)
## [1] 0.005355618
Our distribution estimate is based on 13 observations. Is that enough
for a robust estimate? What if we suddenly hit a year like 2008? How
would that affect our estimate?
Let’s try to include the Velliv data from 2007-2010.
We do this by sampling 13 observations from vmrl.
## m s
## Min. :0.05943 Min. :0.03653
## 1st Qu.:0.06521 1st Qu.:0.06151
## Median :0.06913 Median :0.06847
## Mean :0.07044 Mean :0.06724
## 3rd Qu.:0.07616 3rd Qu.:0.07457
## Max. :0.08687 Max. :0.09149
xiThe fit for mhr has the highest xi value of
all. This suggests right-skew:
If the Law Of Large Numbers holds true, \[\dfrac{\max (X_1^p, ..., X^p)}{\sum_{i=1}^n X_i^p} \rightarrow 0\] for \(n \rightarrow \infty\).
If not, \(X\) doesn’t have a \(p\)’th moment.
See Taleb: The Statistical Consequences Of Fat Tails, p. 192
Comments
(Ignoring
mhr_a…)mhrhas some nice properties:- It has a relatively high
nuvalue of 90, which means it is tending more towards exponential tails than polynomial tails. All other funds havenuvalues close to 3, exceptphrwhich is even worse at close to 2. (Note that for a Gaussian,nuis infinite.)- It has the lowest losing percentage of all simulations, which is better than 1/6 that of
phr.- It has a DAO percentage of 0, which is the same as
mmr, and less thanphr.- Only
phrhas a highermc_m.- It has a smaller
mc_sthan the individual components,vhrandphr.- It has the highest
xiof all fits, suggesting less left skewness. Density plots forvmr,phrandmmrhave an extremely sharp drop, as if an upward limiter has been applied, which corresponds to extremely lowxivalues. The density plot formhris by far the most symmetrical of all the fits. As seen in the section “Compare Gaussian and skewed t-distribution fits”, the other skewed t-distribution fits don’t capture the max observed returns at all.- Only
mmrhas as highermc_min. However, that ofmmris 18 times higher with 62, sommris a clear winner here.- Naturally, it has a
mc_maxsmaller than the individual components,vhrandphr, but ca. 1.5 times higher thenmmr.- All the first 4 moments converge nicely. For all other fits, the 4th moment doesn’t seem to converge.
Taleb, Statistical Consequences Of Fat Tails, p. 97:
“the variance of a finite variance random variable with tail exponent \(< 4\) will be infinite”.
And p. 363:
“The hedging errors for an option portfolio (under a daily revision regime) over 3000 days, un- der a constant volatility Student T with tail exponent \(\alpha = 3\). Technically the errors should not converge in finite time as their distribution has infinite variance.”
mhrwith1e4paths gives a mean of 520, while1e6paths gives a mean of 478 (see under “Many simulations”).